Environmental Engineering Reference
In-Depth Information
Figure 5.10. Dynamics described by model ( 5.17 ) when a is negative ( a
=−
0
.
1,
D
=
10, k 0 =
1). The three panels correspond to t equal to 0, 5, and 35 time units, the
field has 128
128 pixels, and periodic boundary conditions are set. The gray-tone
scale spans the interval [10 3
×
10 3 ]. The initial condition is
,
φ
( r
,
t
=
0)
= ξ u , where
ξ u is a noise uniformly distributed in the interval [10 2
10 2 ].
,
tend to vanish as the system tends to the homogeneous stable state
φ =
0. Notice that
the transient pattern exhibits the periodicity of about 2
π/
k 0 pixels imposed by the
spatial coupling (see Appendix B).
Conversely, when a is positive no steady states exist and the dynamics of
diverge.
However, even in this case the spatial terms in ( 5.17 ) are able to induce patterns
that become more and more pronounced (i.e., with stronger spatial gradients) as the
dynamics diverge (see Fig. 5.11 ). To stabilize the dynamics to a steady state, it is
necessary to introduce a nonlinear term that hampers the dynamics in diverging.
A relatively simple nonlinear term is
φ
3 ; in this case the dynamics turn into the
celebrated Ginzburg-Landau model, with local dynamics expressed as f (
φ
φ
)
=
a
φ
3 ,
φ
∂φ
3
2
k 0 ) 2
t =
a
φ φ
D (
+
φ.
(5.18)
a
, a ] - the diverging effect of the lin-
For small values of
φ
- in the interval [
ear term a
φ
prevails (with a
>
0), and as
φ
increases the stabilizing effect of the
Figure 5.11. Example of the dynamics described by deterministic model ( 5.17 ) when
a is positive ( a
=+
.
=
10, k 0 =
1). The three panels correspond to t equal to
0, 30, and 60 time units, and the gray-tone scale spans the interval [
0
1, D
.
,
.
0
1
0
1]. The
other conditions are as in Fig. 5.10 .         Search WWH ::

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